Q113 / 180 · COMEDK 2024 (Morning Shift) · Mathematics

$\text { If } \dfrac{1}{2}\left(\dfrac{3 x}{5}+4\right) \geq \dfrac{1}{3}(x-6), x \in R \text { then }$

  • A
    $x \in(120, \infty)$
  • B
    $x \in(-\infty, 120)$
  • Correct answer
    C
    $x \in(-\infty, 120]$
  • D
    $x \in[120, \infty)$

Solution

Given the inequality $\dfrac{1}{2}\left(\dfrac{3x}{5} + 4\right) \geq \dfrac{1}{3}(x - 6)$.

Multiplying both sides by $6$ to clear the denominators:

$3\left(\dfrac{3x}{5} + 4\right) \geq 2(x - 6)$

Expanding both sides:

$\dfrac{9x}{5} + 12 \geq 2x - 12$

Rearranging the terms to isolate $x$:

$12 + 12 \geq 2x - \dfrac{9x}{5}$

$24 \geq \dfrac{10x - 9x}{5}$

$24 \geq \dfrac{x}{5}$

Multiplying by $5$:

$120 \geq x$ or $x \leq 120$.

Thus, the solution set is $x \in (-\infty, 120]$.

Answer: $x \in(-\infty, 120]$

Hence, the correct answer is option: 'C'.

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